1. Introduction to Dynamical Mean Field Theory (DMFT) and its Applications to the Electronic Structure of Correlated Materials Zacatecas Mexico PASSI School. Montauk June (2006). G.Kotliar Physics Department Center for Materials Theory Rutgers University. Collaborators: K. Haule (Rutgers), C. Marianetti (Rutgers ) S. Savrasov (UC Davis)

2. References Electronic structure calculations with dynamical mean- field theory: G. Kotliar, S. Savrasov, K. Haule, V. Oudovenko, O. Parcollet, and C. Marianetti, Rev. of Mod. Phys. 78, 000865 (2006). Dynamical Mean Field Theory of Strongly Correlation Fermion Systems and the Limit of Infinite Dimensions: A. Georges, G. Kotliar, W. Krauth, and M. Rozenberg, Rev. of Mod. Phys. 68, 13-125 (1996).Rev. of Mod. Phys. 78, 000865 (2006).Rev. of Mod. Phys. 68, 13-125 (1996) Electronic Structure of Strongly Correlated Materials: Insights from Dynamical Mean Field Theory: Gabriel Kotliar and Dieter Vollhardt, Physics Today 57, 53 (2004).Physics Today 57, 53 (2004).

3. What is a strongly correlated material ?

4. Band Theory: electrons as waves. Landau Fermi Liquid Theory. Electrons in a Solid:the Standard Model Quantitative Tools. Density Functional Theory Kohn Sham (1964) Rigid bands, optical transitions, thermodynamics, transport……… Static Mean Field Theory. 2 Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) n band index, e.g. s, p, d,,f

5. Success story : Density Functional Linear Response Tremendous progress in ab initio modelling of lattice dynamics & electron-phonon interactions has been achieved ( Review: Baroni et.al, Rev. Mod. Phys, 73, 515, 2001 ) (Savrasov, PRB 1996)

6. Kohn Sham reference system Excellent starting point for computation of spectra in perturbation theory in screened Coulomb interaction GW.

7. = W = [ - ] -1 = G - [ - ] GW approximation (Hedin )

8. Kohn Sham Eigenvalues and Eigensates: Excellent starting point for perturbation theory in the screened interactions (Hedin 1965) Self Energy VanShilfgaarde (2005) VanShilfgaarde (2005) 3

9. Strong Correlation Problem:where the standard model fails Fermi Liquid Theory works but parameters can’t be computed in perturbation theory. Fermi Liquid Theory does NOT work. Need new concepts to replace of rigid bands ! Partially filled d and f shells. Competition between kinetic and Coulomb interactions. Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). Non perturbative problem. 4

10. Localization vs Delocalization Strong Correlation Problem A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (fully localized or fully itinerant). Situation realized by applying a control parameters, e.g. pressure. Metal to Insulator Transition. Some materials have several species of electrons, some localized (f ‘s d’s ) some itinerant (sp, spd). OSMT. Heavy Fermions. Introducing carries (electrons or holes) to a Mott insulator. Doping Driven Mott transition.

11. Why is it worthwhile to study correlated electron materials ?

13. Localization vs Delocalization Strong Correlation Problem A large number of compounds with electrons in partially filled shells, are not close to the well understood limits (localized or itinerant). Non perturbative problem. These systems display anomalous behavior (departure from the standard model of solids). Neither LDA or LDA+U or Hartree Fock work well. Dynamical Mean Field Theory: Simplest approach to electronic structure, which interpolates correctly between atoms and bands. Treats QP bands and Hubbard bands.

14. Strongly correlated systems Copper Oxides. High Temperature Superconductivity. Cobaltates Anomalous thermoelectricity. Manganites. Colossal magnetoresistance. Heavy Fermions. Huge quasiparticle masses. 2d Electron gases. Metal to insulator transitions. Lanthanides, Transition Metal Oxides, Multiferroics……………….. 5

15. Basic Questions How does the electron go from being localized to itinerant. How do the physical properties evolve. How to bridge between the microscopic information (atomic positions) and experimental measurements. New concepts, new techniques

16. How do we probe SCES experimentally ?

17. One Particle Spectral Function and Angle Integrated Photoemission Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 Probability of absorbing an electron and transfering energy  =Ei-Ef, and momentum k (1-f(  )) A(  ) M 2 Theory. Compute one particle greens function and use spectral function. e e

18. Spectral Function Photoemission and correlations Probability of removing an electron and transfering energy  =Ei-Ef, and momentum k f(  ) A(  ) M 2 e Angle integrated spectral function 8 a)Weak Correlation b)Strong Correlation

19. Strong Correlation Problem:where the standard model fails Fermi Liquid Theory works but parameters can’t be computed in perturbation theory. Fermi Liquid Theory does NOT work. Need new concepts to replace of rigid bands ! Partially filled d and f shells. Competition between kinetic and Coulomb interactions. Breakdown of the wave picture. Need to incorporate a real space perspective (Mott). Non perturbative problem. 4

20. How do we approach the problem of strongly correlated electron stystems ?

21. Two roads for ab-initio calculation of electronic structure of strongly correlated materials Correlation Functions Total Energies etc. Model Hamiltonian Crystal structure +Atomic positions

22. Strongly correlated systems are usually treated with model Hamiltonians Tight binding form. Eliminate the “irrelevant” high energy degrees of freedom Add effective Coulomb interaction terms.

23. One Band Hubbard model  U/t  Doping  or chemical potential  Frustration (t’/t)  T temperature Mott transition as a function of doping, pressure temperature etc.

24. How do we reduce the many body problem to something tractable ?

25. DMFT Cavity Construction. Happy marriage of atomic and band physics. Reviews: A. Georges G. Kotliar W. Krauth and M. Rozenberg RMP68, 13, 1996 Gabriel Kotliar and Dieter Vollhardt Physics Today 57,(2004). G. Kotliar S. Savrasov K. Haule V. Oudovenko O. Parcollet and C. Marianetti Rev. Mod. Phys. 78, 865 (2006). G. Kotliar and D. Vollhardt Physics 53 Today (2004) Extremize a functional of the local spectra. Local self energy.

26. Mean-Field : Classical vs Quantum Classical case Quantum case Phys. Rev. B 45, 6497 A. Georges, G. Kotliar (1992)

27. Single site DMFT Impurity cavity construction: A. Georges, G. Kotliar, PRB, (1992)] Weiss field

29. Extension to clusters. Cellular DMFT. C-DMFT. G. Kotliar,S.Y. Savrasov, G. Palsson and G. Biroli, Phys. Rev. Lett. 87, 186401 (2001) tˆ(K) is the hopping expressed in the superlattice notations. Other cluster extensions (DCA, nested cluster schemes, PCMDFT ), causality issues, O. Parcollet, G. Biroli and GK cond-matt 0307587 (2003)

30. What is the structure of the DMFT problem ? Embedding and truncation

31. Solving the DMFT equations Wide variety of computational tools (QMC,ED….)Analytical Methods Extension to ordered states. Review: A. Georges, G. Kotliar, W. Krauth and M. Rozenberg Rev. Mod. Phys. 68,13 (1996)]

32. How do we generalize this construction realistic systems ?

33. More general DMFT loop

34. Dynamical Mean Field Theory. Cavity Construction. A. Georges and G. Kotliar PRB 45, 6479 (1992). A(  ) 10

35. A. Georges, G. Kotliar (1992) A(  ) 11

36. Dynamical Mean Field Theory Weiss field is a function. Multiple scales in strongly correlated materials. Exact in the limit of large coordination (Metzner and Vollhardt 89), kinetic and interaction energy compete on equal footing. Immediate extension to real materials DFT+DMFT 12

37. Evolution of the DOS. Theory and experiments 13

38. DMFT Qualitative Phase diagram of a frustrated Hubbard model at integer filling T/W 14

39. Interaction with Experiments. Photoemission Three peak strucure. V2O3:Anomalous transfer of spectral weight M. Rozenberg G. Kotliar H. Kajueter G Thomas D. Rapkine J Honig and P Metcalf Phys. Rev. Lett. 75, 105 (1995) T=170 T=300 15

40. . Photoemission measurements and Theory V2O3 Mo, Denlinger, Kim, Park, Allen, Sekiyama, Yamasaki, Kadono, Suga, Saitoh, Muro, Metcalf, Keller, Held, Eyert, Anisimov, Vollhardt PRL. (2003 ) NiSxSe 1-x Matsuura Watanabe Kim Doniach Shen Thio Bennett (1998) Poteryaev et.al. (to be published) 16

41. How do we solve the impurity model ?

42. Methods of solution : some examples  Iterative perturbation theory. A Georges and G Kotliar PRB 45, 6479 (1992). H Kajueter and G. Kotliar PRL (1996). Interpolative schemes (Oudovenko et.al.)  Exact diag schemes Rozenberg et. al. PRL 72, 2761 (1994)Krauth and Caffarel. PRL 72, 1545 (1994)  Projective method G Moeller et. al. PRL 74 2082 (1995).  NRG R. Bulla PRL 83, 136 (1999)

45. QMC M. Jarrell, PRL 69 (1992) 168, Rozenberg Zhang Kotliar PRL 69, 1236 (1992),A Georges and W Krauth PRL 69, 1240 (1992) M. Rozenberg PRB 55, 4855 (1987). NCA Prushke et. al. (1993). SUNCA K. Haule (2003). Analytic approaches, slave bosons. Analytic treatment near special points.

52. How good is DMFT ?

53. Single site DMFT is exact in the Limit of large lattice coordination Metzner Vollhardt, 89 Muller-Hartmann 89

54. C-DMFT: test in one dimension. (Bolech, Kancharla GK PRB 2003) Gap vs U, Exact solution Lieb and Wu, Ovshinikov Nc=2 CDMFT vs Nc=1

55. N vs mu in one dimensional Hubbard model. Compare 2 site cluster (in exact diag with Nb=8) vs exact Bethe Anzats, [M. Capone C. Castellani M.Civelli and GK (2003)]

56. How do we incorporate the Long Range Coulomb Interactions

57. DMFT Impurity cavity construction

58. How do we merge band theory and DMFT ? How do we extract total energies ?

59. observable of interest is the "local“ Green's functions (spectral function) Currently feasible approximations: LDA+DMFT: Spectral density functional theory (G. Kotliar et.al., RMP 2006). Variation gives st. eq.: Generalized Q. impurity problem!

60. General impurity problem Diagrammatic expansion in terms of hybridization  +Metropolis sampling over the diagrams Exact method: samples all diagrams! Allows correct treatment of multiplets k K.H. Phys. Rev. B 75, 155113 (2007) Exact “QMC” impurity solver, expansion in terms of hybridization P. Werner, Phys. Rev. Lett. 97, 076405 (2006)

61. What are the characteristics of the spectra of a correlated system, in the simplest model ?

62. Pressure Driven Mott transition

63. T/W Phase diagram of a Hubbard model with partial frustration at integer filling. [Rozenberg et. al. PRL 1995] Evolution of the Local Spectra as a function of U,and T. Mott transition driven by transfer of spectral weight Zhang Rozenberg Kotliar PRL (1993).. Mott transition in one band model. Review Georges et.al. RMP 96

64. X.Zhang M. Rozenberg G. Kotliar (PRL 1993) Spectral Evolution at T=0 half filling full frustration

65. Parallel development: Fujimori et.al

66. Evolution of the Spectral Function with Temperature Anomalous transfer of spectral weight connected to the proximity to the Ising Mott endpoint (Kotliar Lange nd Rozenberg Phys. Rev. Lett. 84, 5180 (2000)

67. n=1 Order in Perturbation Theory Order in PT Range of the clusters Basis set size. DMFT GW r site CDMFT l=1 l=2 l=lmax r=1 r=2 n=2 GW+ first vertex correction

68. How do we go directly from structure to physical observables ?

70. Can the various approaches (DMFT, DFT, DFT+U be unified )?

71. Spectral density functional. Effective action construction.e.g Fukuda et.al

73. In practice we need good approximations to the exchange correlation, in DFT LDA. In spectral density functional theory, DMFT. Review: Kotliar et.al. Rev. Mod. Phys. 78, 865 (2006) Kohn Sham equations

74. Different methods differ by the choice of variable a used. DFT Spin and Density FT Spectral Density Functional R. Chitra and G.K Phys. Rev. B 62, 12715 (2000). S. Savrasov and G.K PRB (2005) Phys. Rev. B 62, 12715 (2000).

75. C DMFT extend the notion of “locality”to several unit cells U (and form of dc) are input parameters. EDMFT a=“ Gloc Wloc” Cluster Greens Function and Screened interaction, No input parameters. Recently impelemented and tested for sp systems. Si C …. N. Zein et.al.PRL 96, (2006) 226403 Zein and Antropov PRL 89,126402 Review: Kotliar et.al. Rev. Mod. Phys. 78, 865 (2006) DFT+DMFT

76. What is the ultimate theory, without any external parameters ?

77. Functional formulation. Chitra and Kotliar Phys. Rev. B 62, 12715 (2000) and Phys. Rev.B (2001). Phys. Rev. B 62, 12715 (2000) Ex. Ir>=|R,  > Gloc=G(R , R  ’)  R,R’ ’ Introduce Notion of Local Greens functions, Wloc, Gloc G=Gloc+Gnonloc. Sum of 2PI graphs One can also view as an approximation to an exact Spetral Density Functional of Gloc and Wloc.

78. Classical case Quantum case A. Georges, G. Kotliar (1992) Mean-Field : Classical vs Quantum Easy!!! Hard!!! QMC: J. Hirsch R. Fye (1986) NCA : T. Pruschke and N. Grewe (1989) PT : Yoshida and Yamada (1970) NRG: Wilson (1980) Pruschke et. al Adv. Phys. (1995) Georges et. al RMP (1996) IPT: Georges Kotliar (1992).. QMC: M. Jarrell, (1992), NCA T.Pruschke D. Cox and M. Jarrell (1993), ED:Caffarel Krauth and Rozenberg (1994) Projective method: G Moeller (1995). NRG: R. Bulla et. al. PRL 83, 136 (1999),……………………………………...

80. Full implementation in the context of a a one orbital model. P Sun and G. Kotliar Phys. Rev. B 66, 85120 (2002). After finishing the loop treat the graphs involving Gnonloc Wnonloc in perturbation theory. P.Sun and GK PRL (2004). Related work, Biermann Aersetiwan and Georges PRL 90,086402 (2003). EDMFT loop G. Kotliar and S. Savrasov in New Theoretical Approaches to Strongly Correlated G Systems, A. M. Tsvelik Ed. 2001 Kluwer Academic Publishers. 259-301. cond-mat/0208241 S. Y. Savrasov, G. Kotliar, Phys. Rev. B 69, 245101 (2004)

81. Conclusion DMFT, method under very active development. But there is now a clear formulation (and to large extent implementation) as a fully self consistent, controlled many body approach to solids. It gives good quantitative results for total energies, phonon and photoemission spectra, and transport of materials. Many examples…all over the periodic table. Helpful in developing intuition and qualitative insights in correlated electron materials. With advances in implementation, we will be able to focus on deviations from (cluster) dynamical mean field theory.

82. The Mott transition problem Universal and non universal aspects. Frustration and the success of DMFT. In the phases without long range order, DMFT is valid if T > Jeff. Need frustration to supress it. When T < Jeff LRO sets in. If Tneel is to high it oblitarates the Mott phenomena. t vs U fundamental competition and secondary instabilities.

83. V 2 O 3 under pressure or

84. Schematic DMFT phase diagram of a partially frustrated integered filled Hubbard model.

85. S.-K. Mo et al., Phys. Rev. Lett. 90, 186403 (2003)..

86. Schematic DMFT phase Implications for transport.

89. Material Properties: total energy and phonon spectra